Evolution, 58(10), 2004, pp. 2133–2143
COMPARING STRENGTHS OF DIRECTIONAL SELECTION:
HOW STRONG IS STRONG?
JOE HEREFORD,1,2 THOMAS F. HANSEN,1,3
1 Department
AND
DAVID HOULE1,4
of Biological Science, Florida State University, Tallahassee, Florida 32306-1100
2 E-mail: hereford@bio.fsu.edu
3 E-mail: thomas.hansen@bio.fsu.edu
4 E-mail: dhoule@bio.fsu.edu
Abstract. The fundamental equation in evolutionary quantitative genetics, the Lande equation, describes the response
to directional selection as a product of the additive genetic variance and the selection gradient of trait value on relative
fitness. Comparisons of both genetic variances and selection gradients across traits or populations require standardization, as both are scale dependent. The Lande equation can be standardized in two ways. Standardizing by the
variance of the selected trait yields the response in units of standard deviation as the product of the heritability and
the variance-standardized selection gradient. This standardization conflates selection and variation because the phenotypic variance is a function of the genetic variance. Alternatively, one can standardize the Lande equation using
the trait mean, yielding the proportional response to selection as the product of the squared coefficient of additive
genetic variance and the mean-standardized selection gradient. Mean-standardized selection gradients are particularly
useful for summarizing the strength of selection because the mean-standardized gradient for fitness itself is one, a
convenient benchmark for strong selection. We review published estimates of directional selection in natural populations using mean-standardized selection gradients. Only 38 published studies provided all the necessary information
for calculation of mean-standardized gradients. The median absolute value of multivariate mean-standardized gradients
shows that selection is on average 54% as strong as selection on fitness. Correcting for the upward bias introduced
by taking absolute values lowers the median to 31%, still very strong selection. Such large estimates clearly cannot
be representative of selection on all traits. Some possible sources of overestimation of the strength of selection include
confounding environmental and genotypic effects on fitness, the use of fitness components as proxies for fitness, and
biases in publication or choice of traits to study.
Key words.
Breeder’s equation, elasticity, natural selection, phenotypic selection, selection gradient.
Received March 4, 2004.
Lande’s realization that directional selection could be approximated as a selection gradient (Lande 1979), together
with his proposal of multiple regression as an appropriate
technique for estimation of selection gradients (Lande and
Arnold 1983), resulted in an explosive increase in empirical
estimates of directional selection. As long as some measure
of fitness of individuals is available, gradients can be estimated for any quantifiable trait in any taxon. Selection gradients have been used to describe the pattern of selection for
estimation of adaptive landscapes (e.g., Arnold et al. 2001)
and to test for differences in the pattern of selection in different environments (e.g., by Dudley 1996).
In many of these examples, testing the validity of different
hypotheses requires that the magnitudes of selection in different environments or on different traits be compared. This
comparison necessitates quantifying selection on a common
scale. Almost all such published comparisons have relied on
variance-standardized selection gradients or selection intensities, which measure selection in units of standard deviations. For example, Kingsolver et al. (2001) and Hoekstra et
al. (2001) reviewed gradients and compared them on a variance-standardized scale (see also Endler 1986). While this
is a useful approach to standardization, it has some important
disadvantages. The first of these is that variance-standardized
selection gradients are functions of population variation and
therefore cannot be viewed as descriptors of the fitness function or of the adaptive landscape (sensu Simpson 1944). Second, the variance-standardized gradient offers no clear criteria for determining whether selection is strong. One of the
conclusions of Kingsolver et al.’s (2001) review was that
Accepted July 30, 2004.
selection was generally not strong, but they did not discuss
any criterion for identifying strong selection. Finally, exclusive dependence on one standardization makes it difficult to
separate differences due to the quantity of interest (the selection gradient) from those due to the standardizing factor
itself.
Our purpose in the present paper is twofold. First we call
attention to the advantages of mean-standardized directional
selection gradients (Morgan 1999; van Tienderen 2000).
Mean-standardized gradients better reflect the shape of the
selective surface and provide a natural scale for assessing the
strength of selection, although they are not appropriate for
all traits. Second, we survey published estimates of directional selection, convert them to mean-standardized gradients, and compare the strength of selection on different types
of traits and on different fitness components. This reveals a
picture of the strength of selection very different from that
suggested previously (Hoekstra et al. 2001; Kingsolver et al.
2001).
Standardized Measures of Selection
Lande (1979) paved the way for a new approach to the
measurement of directional selection by decomposing the
response to selection as R 5 Gb, where R is the change in
mean of a (vector) trait, G is the additive genetic variance
matrix, and b is the selection gradient. Following up on a
forgotten suggestion by Pearson, Lande and Arnold (1983)
subsequently showed how the selection gradient could be
estimated as a set of partial regression coefficients from a
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q 2004 The Society for the Study of Evolution. All rights reserved.
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JOE HEREFORD ET AL.
regression of fitness on the trait vector. This method made
it possible to assess the strength of direct selection acting on
a single trait while controlling for selection on correlated
traits. This aspect of Lande’s formulation represented a considerable practical advance, and it has been employed in
many empirical studies.
Another fundamentally important benefit of using Lande’s
equation is less appreciated: It separates a measure of natural
selection (the selection gradient) from the ability of the population to respond. This distinction is easiest to grasp from
the univariate version of the Lande equation:
R 5 s2Ab,
(1)
s2A
where
is the additive genetic variance in the selected trait,
and b is the selection gradient, the regression slope of relative
fitness, w (fitness standardized to a mean of one), on trait
value, z (Lande 1979). The shape of the fitness landscape in
the neighborhood the population inhabits is captured by b,
whereas s2A determines the population’s ability to respond to
selection, the evolvability. As a regression coefficient, b consists of the ratio
b5
COVw·z
,
s 2z
(2)
where COVw·z is the covariance between relative fitness and
trait z, and s2z is the phenotypic variance of trait z (Lande
and Arnold 1983). Note that the phenotypic variance is the
sum of the additive genetic variance and all other sources of
variation, which we label the residual variance (s2R). COVw·z
is also the selection differential, S, the change in average
phenotype within a generation due to selection (Price 1970).
Comparisons of responses to selection are often difficult
on the sole basis of equation (1). The response, R, is in trait
units and s2A in trait units squared, whereas b gives the change
in relative fitness for a unit change in the trait and thus has
units of one per trait. Different traits may be measured in
different units, making direct comparisons of R, s2A, or b
fruitless. Even when traits are measured in the same units,
the means and variances may be very different, again making
comparisons difficult. The obvious solution is to standardize
each quantity to remove the units of measure before comparison. Two such standardization schemes are available.
The most familiar standardization uses the standard deviation of the trait. Equation (1) can be rearranged into the
familiar breeder’s equation as
R5
s 2A
COVw·z 5 h 2 S,
s 2z
(3)
where h2 is the dimensionless narrow-sense heritability and
S is the covariance between the trait and relative fitness.
Standardization is completed by dividing both sides by the
phenotypic standard deviation sz
R
s 2 COVw·z
5 A2
5 h 2bs .
sz
s z sz
(4)
Here, bs 5 COVw·z/sz 5 szb is the variance-standardized
selection gradient (Lande and Arnold 1983), which gives the
change in relative fitness for a change of one standard deviation in the trait mean. It also corresponds to i, the intensity
of selection (Falconer and Mackay 1996), the number of standard deviations by which selection changes the trait mean
within a generation.
This standardization has one major disadvantage. Although
the Lande equation cleanly separates its measure of the fitness
landscape from the properties of the population, the breeder’s
equation does not. The natural measure of evolvability is the
additive genetic variance, s2A, but the standardization factor
sz is itself a function of s2A. Using sz as a yardstick is like
standardizing leg length by dividing by body length; the resulting quantity may be interesting, but is no longer a measure
of leg length. Heritability is thus a singularly misleading
measure of evolvability (Houle 1992; Hansen et al. 2003).
Similarly, multiplying a measure of directional selection by
a function of evolvability yields a hybrid of the two, calling
into question the general significance of bs. Although the
variance-standardized breeder’s equation does have its uses,
as discussed below, separation of evolvability and selection
is not one of them.
A simple example helps to make the problem clear. Imagine two populations that are at the same point on a linear
selective landscape where b 5 0.1 and that have the same
residual variance, s2R 5 10, but where population 1 has a
2
larger s2A than population 2, say s2A.1 5 10, sA.2
5 5. Population 1 will respond more to the given selection pressure,
and the Lande equation clearly shows why: s2A.1 . s2A.2 . However, one would be mislead by interpreting the variance-standardized equation in the same way. The heritability of population 1 (0.5) is greater than that of population 2 (0.33), as
expected, but population 1 has larger sz than population 2.
The two populations therefore differ in the standardization
factor, precisely because of the difference in s2A. This difference leads to a larger estimate of selection in population
1 (bs.1 5 0.45) than in population 2 (bs.2 5 0.39), even though
this example is constructed to have the same strength of
selection. If the additive variances were the same, but the
residual variances differed, the confusion generated would
be complete. In this second case, both the evolvability and
the landscape are the same, but the standardization would
cause both h2 and bs to differ. Thus, what the Lande equation
has clearly separated the standardized breeder’s equation conflates.
Alternatively, the response to selection can be standardized
by the trait mean (Johnson et al. 1955; Houle 1992; Morgan
1999; van Tienderen 2000; Hansen et al. 2003). Recalling
that b has units of one per trait, the directional selection
gradient can be standardized by multiplying by the trait mean
to yield the mean-standardized selection gradient
bm 5 z̄ b 5 z̄
Cov z·w
.
s 2z
(5)
The mean-standardized gradient is the increase in relative
fitness for a proportional change in the trait z and thus is an
elasticity (Caswell 1989; van Tienderen 2000). It is particularly useful that bm is the natural measure of selection in a
mean-standardized version of the Lande equation
R
s2
5 2A z̄ b 5 I Abm ,
z̄
z̄
(6)
where IA is the square of the additive genetic coefficient of
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HOW STRONG IS SELECTION?
variation (CVA 5 sA/z̄). The symbol IA (Houle 1992) was
chosen by analogy to the opportunity for selection, I (Crow
1958; Arnold and Wade 1984). The opportunity for selection
is the squared coefficient of variation of fitness I 5 s2w/W̄2,
and gives the maximum possible response of fitness to selection, that is, the response of fitness if its heritability was
one. Replacing the phenotypic variance with the additive genetic variance to yield IA gives the actual response of fitness
to selection. In the general case where the trait is not fitness
IA measures the opportunity for response to selection.
Although equation (6) is valid for all traits with nonzero
means, the quantities IA and bm only have a natural interpretation on a true ratio scale, where the origin of the scale is
not arbitrary. Examples of true ratio traits are fecundity and
any measure of size, where a value of zero sets a natural
origin for the scale of measurement. An example in which
the origin is arbitrary is measurement of timing of biological
events in calendar days. In such a case, the trait is not true
ratio.
The use of bm has several major advantages as a measure
of the standardized strength of selection. First, the trait mean
is a better standardization factor than the trait variance because neither of the quantities to be compared is a direct
function of the mean. This statement must be qualified, in
that variances are often correlated with means, and the
strength of directional selection will generally depend on the
position on an adaptive landscape. Furthermore, the use of
any standardization factor will introduce a covariance between evolvability and selection.
Second, the strength of selection on fitness provides a useful benchmark from which to judge the strength of directional
selection on other traits. When the selected trait is fitness, it
is easy to show that b 5 1/W̄, so the mean-standardized
selection gradient for fitness (bm 5 W̄ 3 [1/W̄]) is one (Hansen
et al. 2003). This clarifies why I is the opportunity for selection and IA the actual response of fitness to selection. The
bs for fitness is equal to the coefficient of variation in fitness
(bs 5 sw/W̄), rather than a constant. The minimum amount
of directional selection is obviously no directional selection
in either system, which leads to bm 5 bs 5 0. Thus the meanstandardized system provides a benchmark for both strong
selection and the absence of directional selection; in the variance-standardized system the magnitudes of nonzero bs values have so far only been interpreted intuitively (e.g. Endler
1986; Kingsolver et al. 2001).
It is important to realize that bm 5 1 is not a maximum
strength of selection, but a benchmark for selection that everyone can agree is strong. The fitness landscape itself can
of course have areas that are arbitrarily steep, for example,
a threshold below which individuals die and above which
they survive. In such cases, the estimated gradient will depend on the mean and variance in the measured trait. If the
variation is small and the mean is centered on the steepest
part of the landscape, bm can be much greater than one. As
trait variance becomes larger, the rate of change in fitness
over the range of the data will become smaller. An area with
a bm . 1 can only occur when the fitness function is nonlinear
over the entire landscape, as in the threshold case. If the
fitness function is linear with an intercept of zero, then bm
5 1 is indeed a maximum, but this is a special case that is
unlikely to hold for traits that are not fitness components.
Both standardized measures of selection do have maximum
values set by the correlation between fitness and the trait
(Arnold 1986). The correlation coefficient, rw·z, must have a
value between 21 and 1. Starting from the inequality zrw·zz
5 zCOVw·z/swszz # 1, it is straightforward to show that zbmz
# z̄ sw/sz 5 CVw/CVz and zbsz # sw 5 CVW. Note that these
inequalities imply that there are limits to the proportion of
fitness explained by variation in the trait.
The relationships between the dimensionless quantities in
equations (4) and (6) are simple functions of the phenotypic
coefficient of variation, CVz 5 sz/z̄. For the standardized
measures of selection, bs 5 CVz bm. For the measures of
genetic variation, IA 5 h2 (CVz)2. Therefore, if traits have
different phenotypic coefficients of variation, comparisons
based on equation (4) and (6) may lead to different conclusions.
For simplicity, our derivations have all been for the univariate case, where selection is estimated on a single trait.
Both of these standardization systems extend to the multivariate case, where selection is measured on more than one
potentially correlated trait. In such cases, b is a partial regression coefficient (Lande and Arnold 1983), which expresses the projected change in relative fitness for a change
in the trait while all other traits are held constant. The interpretation of standardized partial coefficients and their standardized equivalents is otherwise the same as that of univariate regression coefficients. They differ only in the number
of other traits for which the effects of indirect selection have
been removed—none for univariate estimates and a small
number for multivariate estimates. We expect that in most
cases a large number of unmeasured traits are directly selected and therefore can exert indirect selection on the measured traits.
Note that we have used terminology and symbols slightly
different from those used by Kingsolver et al. (2001). They
reserved the term ‘‘selection gradient’’and the symbol ‘‘b’’
for variance-standardized multiple-regression coefficients,
that is, estimates derived from datasets in which more than
one predictor trait was measured. Kingsolver et al. termed
all estimates of gradients based on univariate regressions (regressions with only a single predictor trait) ‘‘standardized
selection differentials,’’ which they symbolized i, following
Falconer and Mackay (1996). As outlined above, bs 5 i in
the univariate case.
We have not considered the case of nonlinear selection
here. Extension to higher-order terms of the fitness function
is straightforward. For example, quadratic terms can be converted to mean-standardized forms by multiplying by z̄2.
METHODS
We surveyed studies of phenotypic selection published
from 1984 through 2003 in American Journal of Botany,
American Naturalist, Evolution, Ecology, Heredity, International Journal of Plant Sciences, and Journal of Evolutionary
Biology. We used the capabilities of ISI’s Web of Science to
augment our search. Subsequent to our survey, we obtained
the list of studies in Kingsolver et al.’s (2001) review (J. G.
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JOE HEREFORD ET AL.
Kingsolver, pers. comm.) and examined all of the studies on
it. Data were included in our dataset if the study met several
criteria. First, we included studies that estimated univariate
or multivariate selection. In the case of univariate selection,
we included studies that measured selection by direct comparison of trait means before and after selection and those
that calculated a univariate selection gradient by linear regression. These measures comprise the univariate dataset. We
also included studies that estimated multivariate selection
using the standardized selection gradient of Lande and Arnold
(1983). Second, studies had to be conducted in the field and
use natural populations; studies that used phenotypic manipulations were disqualified. Third, so that we could calculate
mean-standardized gradients from regression coefficients and
selection gradients, studies had to include the means and
standard deviations of the trait of interest for the sample used
to calculate the regressions. In a few cases, unpublished
means and variances were obtained directly from the authors.
Finally, we excluded traits measured on scales that were not
true ratio, with the exception of traits measured on a log
scale. To a first approximation, the (univariate) selection gradient on ln(z) is equal to the mean-standardized gradient of
z divided by 1 1 (IP/2), where IP is the mean-standardized
phenotypic variance of z (and IP itself is approximately equal
to the variance of ln[z], assuming a locally linear fitness
function). We located three studies that both measured selection for a trait on a log scale and provided the other necessary statistics.
To calculate mean-standardized selection gradients, we
multiplied regression coefficients of trait on fitness by the
trait mean. For studies that only reported the bs, we multiplied by z̄/sz to obtain bm. Where univariate selection was
measured, we sometimes obtained the selection differentials
directly from the means and variances provided in the articles
and calculated gradients from them. We analyzed the data
for univariate and multivariate selection separately and then
looked for differences between the two measures. We analyzed the data in this way because of the differences in interpretation of univariate and multivariate selection and to
allow comparison of the strength of selection when these
measures were used. Although on average univariate selection measures will be larger than multivariate ones when traits
are correlated, in some cases they will be smaller. To check
the legitimacy of estimates of b, we calculated the correlation
between fitness and the trait as rwz 5 bsz/sw.
We recorded several additional variables for each estimate.
The fitness component used to measure selection was classified as viability, fecundity, or sexual selection. Only survival or survivorship was considered viability, and fecundity
was limited to the number of offspring. Two studies included
a combination of viability and fecundity as the fitness component. These studies were not included in one- and two-way
analyses of differences in selection measures or groupings
by type of fitness component. Female choice, sperm and pollen competition, and male-male competition were all included
as sexual selection. In some studies, different measures of
fitness were regressed on the same sets of traits in the same
population or year. For these studies, we only included the
estimates of selection that used the most comprehensive measure of fitness, to make sure that the same episodes of se-
lection were not repeatedly analyzed. The traits involved
were coded as morphological or life-history traits. Only life
span, timing of reproduction, and number of flowers were
considered life-history traits. We did not include characters
made up of dates of events such as flowering or germination
because dates are not on a ratio scale. One study included
behavioral traits, but these were not included in analyses of
type of trait because they were from a single study and comprised only four estimates. For each estimate we recorded
the sample size, the level of statistical significance, and the
standard error of the estimate where they were available.
For our purposes, the signs of selection estimates are arbitrary, so we analyzed the absolute values of the estimates.
Two-way analysis of variance was performed in the SAS
procedure GLM (SAS, ver. 8.1; SAS Institute, Cary, NC)
using type of trait and fitness components as independent
variables and the selection measure as the dependent variable.
Significance of these analyses was judged relative to the distribution of F-ratios from randomized datasets. The estimates
and sample sizes were randomly shuffled 1000 times relative
to the independent variables, preserving the pairing of estimate and sample size. Each randomized dataset was analyzed
with GLM and the F-ratio obtained. The SAS programs of
Cassell (2002) were used to perform the randomization tests,
after some modification. These tests are all approximate, as
they all treat each selection estimate as independent. In fact,
each set of estimates for the same population at the same
time are correlated with each other, but because the vast
majority of comparisons are between estimates from different
populations and studies, we suspect that this inaccuracy is
small.
Correcting for Bias in Absolute Values
When comparing the strengths of selection on different
traits and in different studies, we naturally wanted to compare
the absolute value of selection. Unfortunately the absolute
values of selection gradients are biased upward whenever the
confidence limits of the estimate overlap zero. To see this,
imagine studying a set of traits that are not under directional
selection. With estimation error, the estimates of b will not
equal zero, although they can be expected to average zero.
When the absolute values of the resulting standardized estimates are calculated, they will average more than zero, as
they must in this case be nonnegative, giving a misleading
impression of the strength of selection. Omitting estimates
whose confidence intervals overlap zero is not a solution, as
doing so would also cause an upward bias in the estimates.
To correct for this bias we began with the assumption that
the estimated selection slope, b, is normally distributed with
a mean, b, equal to the true value of the selection gradient
and a variance, s2, that we assume to be equal to the square
of the reported standard error. It therefore follows that the
variable zbz/s follows a chi distribution (i.e., the distribution
of the square root of a chi-square–distributed variable) with
one degree of freedom and noncentrality parameter (b/s)2
(Evans et al. 2000). On this basis we can compute the expected value of zbz as
2
2(b/s) 2
b
zb z
E[ z b z ] 5 s
Exp
1
Erf
, (7)
p
2
s
sÏ2
5!
[ ]
) ) 1 26
2137
HOW STRONG IS SELECTION?
FIG. 1. Relative bias of absolute values of selection estimates,
shown as a percentage of the estimate, as a function of the relative
error of the regression slope (the standard error of the estimate as
a percentage of the estimate).
where Erf[x] 5 (2/Ïp ) #x0 Exp[2t2] dt is the error function.
If b 5 0, then E[zbz] 5 sÏ2/p . If the relative error, s/zbz, is
small, E[zbz] ø b.
Figure 1 shows the relative bias (bias measured in percentage of the mean) plotted against the relative error of the
regression slope. The bias is very large when the relative
error exceeds 100% (i.e., when the standard error is as large
as the estimated slope) but decreases rapidly at lower values.
When the relative error is 50%, the bias is less than 1% of
the mean, and it decreases very rapidly below that. Therefore,
as a rule of thumb, the bias is very small for selection gradients that are significantly different from zero.
Many of the estimated selection gradients in our sample
have relative errors in excess of 100%, and in many cases
these estimates are quite large. In these instances the bias
has a genuine impact on our results. As a partial correction
for the bias, we substituted the estimated b for b in equation
(7) to get an estimate of the bias as
bias 5 E[zbz] 2 zbz
(8)
for those studies that gave the necessary standard error. This
bias is underestimated, as the true b is smaller in expectation
than the estimated b used to derive the bias. This remaining
bias cannot be removed in any obvious way, as further iteration will not converge and may lead to an overestimate
of the bias. Before computing standardized gradients, we corrected the absolute values of the selection gradients by subtracting the bias as computed by equation (8). In cases where
the bias exceeded 100% of the mean, we adjusted the absolute
selection gradient to zero. This procedure can be justified by
Bayesian reasoning, as it reasonable to use a prior with zero
weight on negative selection strength.
RESULTS
We were able to locate only 38 studies that included all
the information necessary for calculation of mean-standardized gradients and met our other criteria for inclusion (see
Appendix available online at http://dx.doi.org/10.1554/
04.147.1.s1). They furnished a total of 340 multivariate es-
FIG. 2. Absolute values of the correlations between fitness measures and trait values. Top, partial correlations based on multivariate
estimates; bottom, correlations based on univariate selection estimates.
timates of selection (multivariate defined as estimates derived
by multiple regression with at least two traits) and 240 univariate estimates. This is less than a third of the number that
Kingsolver et al. (2001) reported, despite our inclusion of
more recent work. The difference arises primarily because
published estimates often do not include the means and variances necessary for calculation of mean-standardized gradients, bm.
For 81% of the multivariate estimates and 62% of the
univariate estimates, the variance in relative fitness was also
published, allowing us to calculate the correlations between
fitness and the trait, rwz. Figure 2 shows the absolute values
of these correlations or partial correlations. The median multivariate rwz was 0.11, and the median univariate rwz was 0.23.
Six univariate rwz exceeded the maximum possible value of
one, as did nine of the multivariate estimates. These results
could only be due to errors in calculating selection estimates,
so we eliminated these estimates before further analysis. The
square of rwz is the proportion of variation in fitness explained
by each trait.
Mean-Standardized Gradients
The distribution of absolute values of mean-standardized
selection gradients, bm, is shown in Figure 3. Both the uni-
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JOE HEREFORD ET AL.
FIG. 3. Distribution of absolute values of mean-standardized gradients (zbmz). Top, multivariate zbmz; bottom, univariate zbmz. Large
panels plot the data on an arithmetic scale; insets plot the same
data on a log10 scale. Open bars denote estimates that are not significantly different from zero; closed bars denote significant values.
variate and multivariate distributions had modes near zero
and very long tails of large estimates. The median uncorrected
multivariate bm was 0.54. Therefore, more than half the traits
studied were under selection at least half as strong as selection on fitness itself. Twenty-five percent of the multivariate
bm were greater than 1.34; the maximum had an absolute
value of 25.18. Univariate bm were substantially larger, with
a median of 1.45, suggesting selection much stronger than
the selection on fitness; the 75% quantile was 3.72. To get
a better idea of the effect of controlling for at least some
correlated traits, we compared estimates for which both a
univariate and a multivariate estimate were available. The
multivariate and univariate medians were similar (1.00 and
1.06, respectively).
The very large sizes of the median bm raise the possibility
that the estimates were dominated by estimates with large
standard errors and upwardly biased absolute values. Correcting for bias due to use of absolute values decreases the
multivariate bm substantially. Taking only studies where the
standard errors of the gradients are available, the median
multivariate bm was 0.50 before correction (close to that of
FIG. 4. Plots of sample size versus selection gradient. The reference line is plotted at zero. Top, plot of sample size and multivariate bm; bottom, that of sample size and multivariate bs.
all estimates) and only 0.31 after correction, a decrease of
38%. The vast majority of the difference was due to estimates
that were not significantly different from zero (shown as open
symbols in Fig. 3). The long tail of extremely large values,
most of which are significant, was less affected by the bias
correction. The larger quantiles of the bias-corrected distribution were thus less affected; the 75% quantile for example
was 1.04, only 22% less than the value based on all the
multivariate estimates. Less than 20% of the univariate estimates had the standard errors necessary for calculation of
a bias correction, and these estimates were substantially
smaller than the other univariate estimates, having a median
value of only 0.48. Bias correction resulted in a much smaller
reduction of only 6%.
Another window on this phenomenon is provided by funnel
plots of multivariate bm values as a function of sample size,
shown in the upper part of Figure 4. The magnitude of estimates is reduced to some extent for estimates of large sample size, but many of the estimates for the largest sample
sizes still exceed one. Consistent with this result, the samplesize-weighted mean multivariate bm is 0.97, considerably
larger than the median.
Examination of Table 1 suggests that life-history traits
have values of bm almost twice as large as those of morphological traits in the multivariate dataset, but the difference is
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HOW STRONG IS SELECTION?
TABLE 1. Median absolute value of mean-standardized selection gradients, zbmz, categorized by the type of trait studied (life history or
morphological) and by the fitness measure used (fecundity, mating success, or viability). Sample sizes are given in parentheses after
each estimate.
Raw zb mz
Mult.1
Life history
Morphological
Fecundity
Mating
Viability
All estimates
0.86
0.53
0.38
0.76
0.48
0.54
(42)
(296)
(191)
(107)
(24)
(340)
Univ.2
0.28
1.90
0.74
1.93
0.70
1.45
(30)
(206)
(54)
(146)
(32)
(240)
Bias corrected zb mz
Combined3
0.43
0.72
0.37
1.59
2.83
0.68
(61)
(383)
(215)
(177)
(38)
(448)
Mult.
1.19
0.28
0.25
0.39
0.49
0.31
(13)
(189)
(128)
(56)
(17)
(193)
Univ.
0.18
0.89
0.31
0.75
5.70
0.48
(21)
(28)
(27)
(21)
(1)
(49)
Combined
0.66
0.29
0.23
0.39
0.57
0.30
(33)
(190)
(148)
(56)
(18)
(227)
1
Denotes estimates based on multiple regressions.
Univariate estimates.
3 The combined dataset uses a single estimate for each population/trait combination available, whether univariate or multivariate. If both multivariate
and univariate estimates are available for the same trait and population, the multivariate estimate with the most covariate traits is used.
2
almost fourfold in the opposite direction in the univariate
dataset. Sexual (mating) selection appears stronger than fecundity or viability selection, whereas viability selection
seems strongest in the combined dataset. Despite the large
sizes of some of the differences, our randomization tests sug-
gest that none of these differences are statistically significant,
except with regard to trait type in the combined dataset. There
morphological traits had significantly larger elasticities than
life-history traits (P 5 0.03).
Variance-Standardized Gradients
The distribution of variance-standardized selection gradients, bs, is shown in Figure 5. Like the bm, selection gradients
have modes at zero and long tails of large values. The median
values across the dataset are shown in Table 2. The median
uncorrected value of multivariate bs was 0.09, and the 75%
quantile was 0.29. This median is substantially lower than
the comparable value of 0.16 reported by Kingsolver et al.
(2001). Our requirement for more published information resulted in a sample of studies showing weaker selection. The
univariate estimates had a median of 0.19. When the 95 observations with both univariate and multivariate estimates
were compared directly, the univariate median was only
slightly lower (0.17 vs. 0.21). Correcting for bias decreased
the multivariate gradients by 33%, from 0.09 to 0.06. Bias
correction of the relatively small number of univariate bs
with the necessary data resulted in a median of 0.15.
A funnel plot of multivariate bs values is shown in the
lower part of Figure 4. There are many larger estimates with
large sample sizes. Consistent with this, the sample-sizeweighted mean bs is 0.16, larger than the median estimate
(0.09).
Variance-standardized gradients showed inconsistent relationships with trait type and fitness measure depending on
whether the multivariate, univariate, or combined datasets
were used, as was the case for the bm above. The randomization tests showed that the interaction between trait type
and fitness measure was just significant at P 5 0.03 for the
uncorrected multivariate dataset. No other effects were significant across all six data partitions.
Comparison of Mean- and Variance-Standardized Gradients
FIG. 5. Distribution of absolute values of variance-standardized
selection gradients, zbsz. Top, multivariate zbsz; bottom, univariate
zbsz. Large panels plot the data on an arithmetic scale; insets plot
the same data on a log10 scale. Open bars denote estimates that are
not significantly different from zero; closed bars denote significant
values.
The relationship between multivariate bm and bs is shown
in Figure 6. The values of bm and bs had modest Pearson’s
correlations that were significantly different from both zero
and one by bootstrapping (rp 5 0.63, N 5 340, upper 95%
limit 0.70, lower 95% limit 0.57) but larger rank correlation
(rs 5 0.87, upper 95% limit 0.89, lower 95% limit 0.84). The
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JOE HEREFORD ET AL.
TABLE 2. Median absolute value of variance-standardized selection gradients, zbsz, categorized by the type of trait studied and by the
fitness measure used. Presentation is as in Table 1.
Raw zb sz
Mult.
Life history
Morphological
Fecundity
Mating
Viability
All estimates
0.25
0.09
0.08
0.18
0.10
0.09
(42)
(296)
(191)
(107)
(24)
(340)
Bias corrected zb sz
Univ.
0.30
0.19
0.21
0.19
0.07
0.19
(30)
(206)
(54)
(146)
(32)
(240)
Combined
0.18
0.11
0.08
0.19
0.19
0.12
correlations of the 240 univariate estimates were somewhat
lower but still different from both zero and one (rp 5 0.28,
upper 95% limit 0.38, lower 95% limit 0.20; rs 5 0.71, upper
95% limit 0.75, lower 95% limit 0.63).
DISCUSSION
Standardized Measures of Selection
Our purpose in the present paper is twofold. First we call
attention to two standardized forms of the Lande equation
for the response to selection (eq. 1, R 5 s2A b), one based on
standardizing the fundamental parameters using the trait variance (the breeder’s equation, R/s 5 h2bs) and the other on
standardizing by the trait mean (R/z̄ 5 IA bm). Second, we
review estimates of directional selection using the mean-standardized selection gradient, bm. Mean-standardized gradients
give proportional changes in fitness for a proportional change
in trait value and are thus elasticities. Our review makes clear
that, if these estimates are taken at face value, the average
strength of directional selection observed in nature is extremely strong. These estimates are in aggregate so large that
we doubt that they can be considered typical for reasons that
we discuss below.
Previous reviews of the strength of directional selection
have used the variance-standardized selection gradient, bs,
FIG. 6. Scatter plot of the relationship between multivariate bm
and multivariate bs, grouped by type of trait and fitness measure.
Solid circles, fecundity selection on life-history traits; open circles,
fecundity selection on morphological traits; solid triangles, sexual
selection on life-history traits; open triangles, sexual selection on
morphological traits; solid squares, viability selection on life-history traits; open squares, viability selection on morphological traits.
(61)
(383)
(215)
(177)
(38)
(448)
Mult.
0.35
0.06
0.06
0.15
0.11
0.06
(13)
(190)
(128)
(56)
(18)
(207)
Univ.
0.09
0.17
0.14
0.15
0.18
0.15
(21)
(28)
(27)
(21)
(1)
(49)
Combined
0.30
0.06
0.06
0.15
0.14
0.07
(33)
(191)
(148)
(56)
(19)
(228)
the measure of directional selection appropriate to the breeder’s equation. We outlined why we believe that bs is not an
appropriate general measure of the strength of selection, although it is informative in other ways, as demonstrated by
the reviews of Kingsolver et al. (2001; see also Hoekstra et
al. 2001). For example, if one is interested in predicting the
number of standard deviations that a trait will change given
an estimate of selection, bs is ideal. The variance-standardized selection gradient is also directly related to statistical
power. The availability of bs values allowed Kingsolver et
al. (2001) to show that the power of most studies to estimate
typical strengths of selection was very low.
What the variance-standardized selection gradient does
measure is the rate of change in fitness per standard deviation
change in the trait. Thus, the higher bs is, the bigger the
fitness difference predicted between extreme individuals in
the population and the higher the selection load. We therefore
suggest that bs be referred to as the ‘‘population strength’’
of selection to emphasize its applicability to a particular population on a selective landscape.
In contrast, mean-standardized gradients, bm, tell us about
the fitness consequences of proportional changes in trait
means, without regard to the variation within the population.
This procedure makes biological sense for traits measured
on a true ratio scale, where zero is a meaningful limit to
phenotypic value, denoting a true absence. A key advantage
of mean-standardized gradients is that the bm of relative fitness is one, a useful benchmark from which to judge the
strength of directional selection on other traits. Mean-standardized gradients are thus likely to be on a biologically
meaningful scale for any true ratio trait. We suggest that bm
be referred to as the ‘‘landscape strength’’ of selection to
emphasize that it measures the slope of the fitness landscape
at the population mean in proportional units.
Mean-standardized gradients have some clear advantages.
One is ease of interpretation, as we have emphasized. Second,
selection on characters that have drastically different distributions such as multinomially distributed characters, which
might take on only a few values, and continuous characters,
which can take on an infinite number of values, can be compared on a proportional scale. Comparing selection on these
types of characters is more difficult in units of standard deviation because the variances of these distributions are drastically different. Third, means are more easily estimated than
variances, so the error introduced by standardizations themselves is reduced. Mean-standardized gradients can be interpreted as elasticities (van Tienderen 2000), linking them to
HOW STRONG IS SELECTION?
a large literature on demographic elasticities (Caswell 1989).
Finally, mean-standardized gradients provide a link between
selection and evolvability, measured as mean-scaled additive
genetic variances (IA; Houle 1992; Hansen et al. 2003), all
on compatible scales.
The relative usefulness of the two standardizations rests
in part on the investigator’s intuition about whether a change
in trait means or trait variances is more biologically significant or more likely, for example, due to environmental
changes. Consider a series of populations subject to the same
simple linear fitness function, w(z) 5 a 1 bz. On the one
hand, populations could be compared in which the mean is
the same but phenotypic variances differ. In this case, populations with small variances will have low bs values, which
might be taken to mean that the fitness function itself differs,
when in fact only the population variance does. On the other
hand, if populations are compared that have different means
but the same variance, populations with smaller means will
have lower bm, again perhaps giving the impression of differences in the fitness function when none exists. Any standardization scheme depends on both the properties of interest
(in this case the strength of selection) and the differences in
the standard used.
The above scenario is of course unrealistic in many respects. Real differences between populations are likely to
include differences in both means and variances, as these are
often related to each other. Similarly, fitness functions will
rarely be linear, so the slope of the fitness function is likely
to change with population means. No simple standardization
scheme will serve all purposes or address all problems. Simultaneous use of more than one approach helps to guard
against overinterpretation of results based on any one measure.
Strength of Selection
Our review of mean-standardized gradients, bm, is notable
for the extremely strong selection observed overall. The median bm of 0.54 means that doubling the value of the average
trait increases fitness by 54%. Judged against the scale of the
strength of selection on fitness, it is also 54% as strong as
selection on fitness. This median is biased upward because
of our dependence on the absolute value of bm as a summary
measure of selection. Even when this bias is corrected to the
extent possible, the median bm is still 0.28, or 28% of the
strength of selection on fitness.
This conclusion is in sharp contrast to that of Kingsolver
et al. (2001; see also Hoekstra et al. 2001), who concluded
that directional selection on most traits is weak based on their
summary of variance-standardized selection gradients, bs.
Kingsolver et al.’s conclusion that selection is usually weak
is not explicitly justified by them, but close reading of their
paper suggests that it is based at least partially on the fact
that the estimates have a mode at zero. The difference between their conclusions and ours does not result from our
use of a different and smaller set of studies. The median bs
that we observed was 0.09 (before bias correction), almost
50% less than the value of 0.16 reported by Kingsolver et
al. This difference implies that, had we been able to include
2141
all the estimates of selection in their review, our median bm
would have been even larger.
Consideration of bs estimates in light of what they actually
measure—the rate of change in fitness within the range of
the population—suggests that they can also be seen as large.
Most modest-sized populations range over at least four standard deviations, so it is reasonable to compare the fitness of
individuals this many standard deviations apart. With fitness
standardized to a mean of one, the median bs of 0.16 indicates
that these extreme individuals will differ by 0.64 in relative
fitness. This means that the least fit individuals will be at
least 50% less fit relative to the best phenotype in the population, and that mean fitness is reduced by about 25%. Our
best estimate of the median bs is considerably lower at 0.06,
but it still indicates a range of fitnesses of 20% due to the
typical trait and a reduction in mean fitness due to each trait
of 11%.
These estimates of selection are so large that they call into
question the representativeness or meaning of this sample of
estimates. The typical study in our sample addressed only a
modest number of traits, averaging 3.5. If these are considered to be a random sample of traits, then it is very difficult
to see how a population could sustain directional selection
of this magnitude on even 50 such traits, much less the nearly
infinite number of characters into which an individual phenotype could be decomposed. This paradox could be resolved
in several nonexclusive ways. First, the choice of study systems may be biased toward those in which strong directional
selection is occurring. Second, the choice of traits for study
might be skewed toward those under the strongest selection.
Third, there may be publication bias. Fourth, the dimensionality of the phenotype may be low enough that selection acts
on only a small number of causally independent traits. Fifth,
the estimates that are obtained may be biased upward as a
result of covariance between environment and phenotype.
Sixth, the estimates are based on fitness components that may
be poorly correlated with fitness. If any of these potential
explanations is correct, one cannot directly infer a typical
strength of selection from the available data.
The first three potential explanations involve a preference
on the part of investigators for studying (or publishing) systems and traits under strong selection and suggest that a randomly chosen trait in a randomly chosen population would
typically be under much weaker directional selection. We
attempted to determine which traits were chosen for study
‘‘at random’’ and which because of some a priori expectation
that they would be under selection. In many cases, an a priori
expectation was clearly evident. For example, many studies
of sexual selection looked at ornaments that clearly imply an
expectation of mate choice. In the majority of studies, however, authors did not clearly state whether they considered
some a priori prediction. A few studies estimated selection
on a very large number of traits, so the investigators seem
very unlikely to have had an expectation about selection on
each one. It seemed likely to us that a preference for studying
strong selection is at least part of the explanation for the very
high average strength of selection, but contrary to this expectation, Kruskal-Wallis tests revealed no significant differences based on the a priori expectations expressed in the
2142
JOE HEREFORD ET AL.
published papers. This result may reflect a failure to express
those expectations rather than a lack of investigator choice.
Kingsolver et al. (2001) reported some modest evidence
for publication bias for linear selection gradients, that is a
failure to publish nonsignificant results. The low power of
most studies of phenotypic selection (Kingsolver et al. 2001;
Hersch and Phillips 2004) certainly gives a great deal of
opportunity for such bias to operate. However, the vast majority of the estimates in our dataset are from studies that
estimate selection on many traits or in many populations,
lessening the opportunity for the significance of any particular estimate to influence its publication.
The fourth possible explanation—that effectively only a
few axes of independent phenotypic variation are important
to organism fitness—implies that even a fairly haphazard
choice of traits to measure could capture variation along all
these axes without study of large numbers of traits. The multiple-regression approach would then spread the impact of
selection over those traits that indicate each axis. In this case,
we predict that the total strength of directional selection
would continue to fall as more and more traits were added.
In an extremely simple hypothetical example, all real selection might be due to variation in organism size. The size of
almost any morphological character would then detect this
selection, but if more and more morphological traits were
measured, on average, less and less selection would appear
to affect each one. There is evidence that this occurs between
estimates based on one trait (the unvariate estimates), and
those based on more than one (Tables 1, 2). Within the multivariate studies, however, there is no evidence for this effect.
The rank correlations of number of traits in a study with
strength of selection are actually positive although not significantly different from zero (0.11 for bm and 0.20 for bs).
The fifth potential explanation for the very strong selection
detected is that the estimates are themselves biased upward.
A likely source of such a bias is environmentally induced
covariance between the trait and fitness (Rausher 1992). Environmental covariance can artificially increase estimates of
phenotypic selection in the following way: If sites or territories vary in their suitability to support growth, and therefore
in the fitnesses of individuals that inhabit them, the environment and fitness covary, which can create covariance between
the phenotype and fitness that will be detected as directional
selection by the methods of Lande and Arnold (1983). There
is experimental evidence that such biases can be substantial
(Stinchcombe et al. 2002; Winn 2004). Methods for eliminating environmental covariance have been described (Price
et al. 1988; Rausher 1992; Scheiner et al. 2002), but these
are neither perfect nor commonly used. Environmental covariance is likely to have influenced some of the estimates
of selection reviewed here.
Finally, we note that selection is measured relative to a
fitness component and not to true fitness. Fitness components,
such as viability or fertility, represent only a part of the total
selection and are usually measured in ways that cover only
a small part of the life history of the organism. Furthermore,
there are theoretical reasons to believe that fitness components are sometimes negatively genetically correlated with
each other (Robertson 1955; Charlesworth 1990; Reeve et al.
1990; Houle 1991), such that the effects of selection gen-
erated through one component are reduced or nullified by
selection on other components. The empirical evidence is,
however, equivocal, and on the phenotypic level, positive
correlations may dominate (for reviews see Houle 1991; Roff
1992; Stearns 1992). In any event, the correlation between
fitness components and true fitness is far from perfect, and
if estimated mean-standardized gradients or intensities are to
be interpreted as measures of total selection acting on the
trait, rather than as measures of the selection generated by
the individual fitness component in question, they are severe
overestimates. To get a feel for this bias, we may use the
fact that elasticities follow a chain rule; the elasticity with
respect to total fitness is the product of the elasticity with
respect to the fitness component multiplied with the elasticity
of the fitness component with respect to total fitness:
d ln W
5
d ln z
O ]] lnln FW dd lnln Fz ,
i
i
(9)
i
where W is fitness, Fi are fitness components, z is the trait,
and the sum is over all fitness components affected by the
trait. The fitness components used in the studies we have
reviewed may differ considerably in their effect on total fitness.
In their review of elasticities of avian life-history variables
with respect to population growth rate (l, computed from
life tables), which we can take as a better estimator of total
fitness, Sæther and Bakke (2000) found that the elasticity of
adult survival to l was in the range 0.35 to 0.95, whereas
the elasticity of fecundity to l ranged from 0.05 to 0.65.
Multiplying our median bias-corrected bm (for combined multivariate and univariate estimates) with respect to viability,
0.57, by the approximate mean elasticity of viability with
respect to l from Sæther and Bakke (0.6, estimated from
their fig. 1) produces an average bm with respect to l of
approximately 0.40. More dramatically, multiplying our median bias-corrected bm for fecundity, 0.23, by Sæther and
Bakke’s value of 0.25 produces a bm of 0.06 with respect to
l. These numbers tell us that typical selection on the trait,
as mediated through viability and fecundity, respectively,
would be 40% and 6% as strong as selection on fitness. Although this selection can still be characterized as moderately
strong, the numbers seem much more realistic than the naive,
uncorrected numbers of 283% and 37%, respectively, that
we get if both the bias from use of fitness components and
the estimation bias were ignored.
On the other hand, if each selective episode in the life
cycle results in selective death, the load induced by selection
may still be quite substantial, even when net selection is
weak. This is the sort of selection that favors a plastic response to environmental conditions. Clearly, the task of inferring a typical strength of selection is fraught with pitfalls.
Careful consideration must be given to the choice of study
populations, traits to study, the covariances among traits, and
the measure of fitness, as well as to the causes of fitness
variation.
Finally, we note that the data gathered in this review relied
on the publication of trait means, variances, and clearly described methods of data analysis. Most published studies are
lacking some of these fundamental details. We urge authors,
reviewers, and journal editors to insist on the publication of
HOW STRONG IS SELECTION?
these important details along with the main conclusions of a
study. As is the case in so many other areas of evolution and
ecology, haphazard standards of reporting and the lack of
attention to the meaning of estimates greatly impairs our
ability to generalize on the basis of past work.
ACKNOWLEDGMENTS
We thank D. J. Fairbairn, J. Maad, and A. A. Winn for
sharing unpublished data; J. G. Kingsolver for furnishing his
compilation of gradient estimates; M. Morgan for making
available his unpublished manuscript; and C. Fenster, J. G.
Kingsolver, F. Knapczyk, M. Morgan, J. Travis, A. A. Winn,
and two anonymous reviewers for comments on the manuscript. This work was supported in part by National Science
Foundation grant DEB-0129219 to DH.
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Corresponding Editor: C. Fenster